Biomechanics, motor control and dynamic models of the soft limbs of the octopus and other cephalopods

An increasing number of studies are currently modeling and constructing bio-inspired soft-robotic systems that are more flexible and versatile than their articulated counterparts and more suitable for performing unconstrained free motions and interaction tasks. Many of these studies have been inspired by investigations in flexible hyper-redundant cephalopods, particularly the octopus and the squid. Indeed, the octopus has emerged as a critical source of inspiration for designing the control principles and strategies underlying motion and force production in soft robotic limbs (Chang et al., 2020a; Della Santina et al., 2021; Gazzola et al., 2018; Ian et al., 2005; Laschi et al., 2016; Li et al., 2022; Ricotti et al., 2017; Rus and Tolley, 2015; Sfakiotakis et al., 2013, 2015; Sivitilli et al., 2022; Thuruthel et al., 2018; Trimmer, 2017, 2020; Trivedi et al., 2008; Walker et al., 2005; Zhang et al., 2019).

Cephalopod arms lack a rigid skeleton; the muscles are organized and neuronally controlled to supply both ‘skeletal-like’ support and the dynamic components necessary for movement. The body is almost entirely composed of muscular tissue, thus transmitting pressure within and among muscular components. Animal bodies and organs with this type of morphology, in which the muscular tissue acts as an effector and as a supporting structure, are called ‘muscular hydrostats’. Muscular hydrostats other than cephalopod arms and tentacles are the elephant trunk and vertebrate tongues (Kandhari et al., 2018; Kier, 2012; Kier and Smith, 1985; Ludwig and Trimmer, 2020). The octopus arm is a particularly interesting muscular hydrostat because of the flexibility, agility and versatility manifested in its ability to elongate, shorten, bend and twist at any point along the arm and perform complex tasks (Fiorito et al., 1990; Kennedy et al., 2020; Levy and Hochner, 2017; Nesher et al., 2020; Wells and Wells, 1957).

Here, we review research on morphology, biomechanics, neural control, motion repertoire and dynamic modeling of octopus arm movements, also comparing the octopus with other muscular hydrostats. We show how many of the hypotheses and speculations have been based on experiments employing a relatively limited range of methodologies and are thus incomplete. It is yet unclear, for example, to what extent arm movements are generated by peripheral activation of specific muscle groups together with central control of muscle synergies and to what extent passive and neuronally triggered control modes are differentially employed during specific motions. These aspects are highly pertinent in both computational models and robotic implementations.

We describe research results in parallel with the methodologies employed, offering a more comprehensive background to the findings. This reveals the existing gaps, puzzling questions, challenges and directions needed to progress significantly in understanding the biology, computational aspects and the neuromuscular control of octopus arm dynamics. Through a new understanding of the biomechanics, morphology and neural motor control of soft limbs and bodies, we believe that a novel approach to the manufacturing and control of soft actuators may emerge, reducing the computational complexity associated with their control and their interactions with the environment.

Studies of muscular hydrostats have a long history in the Journal of Experimental Biology, including prominent comparative investigations by Kier, Van Leeuwen, Chiel, Trimmer, Nishikawa and earlier by Trueman and several others. Kier and colleagues described in detail the fine and gross morphology, biomechanics and biochemistry of muscular hydrostats, ranging from vertebrate tongues and trunks to invertebrates' bodies and muscular organs (Kier and Thompson, 2003; Kier, 2012, 2016; Kier and Smith, 1985, 1990; Kier and Stella, 2007; Kurth et al., 2014; Taylor-Burt et al., 2018; Taylor and Kier, 2003; Van Leeuwen and Kier, 1997). Other extensive studies provided the necessary groundwork for further research on muscular hydrostats (Bone and Trueman, 1983; Chiel and Beer, 1997; Chiel et al., 1992, 2009; Hodgson and Trueman, 1981; Kristan et al., 2000; Metallo et al., 2020; Nishikawa, 2020; Schulz et al., 2022; Ting and Chiel, 2017; Trueman, 1968; Trueman and Brown, 1985; Trueman and Wong, 1987; Van Leeuwen and Kier, 1997; Vaughan et al., 2018). Yet, many questions remain open.

Cephalopod arm musculature is largely composed of longitudinal and transverse muscle layers with several thin layers of oblique muscles (for a detailed description, see Fossati et al., 2011; Kier, 2016) (Fig. 1A). Differences in the reciprocal arrangement and functions of the octopus arm muscles have been revealed by functional, morphological and simulation studies (Di Clemente et al., 2021; Kennedy et al., 2020; Van Leeuwen and Kier, 1997; Zullo et al., 2022). In addition, the muscle fibers composing cephalopod arms and tentacles show clear differences, which should be considered when modeling such structures.

Electron microscopy has revealed that the arm's longitudinal and transverse muscles are composed of obliquely striated muscle cells, typical of many muscular hydrostats (Kier, 2012; Kier and Smith, 1985). Oblique striation may allow high force output over extraordinary length ranges (i.e. super-elongation) (Toida et al., 1975; Thompson et al., 2023), but ongoing studies from Kier's group suggest a new interpretation of the function of oblique striation. A recent comparative investigation of force-length relationships in obliquely striated muscle suggests that oblique striation has evolved to increase the flexibility of the force-length relationship (Thompson et al., 2023).

Transverse muscle fibers in decapod tentacles are cross-striated with much shorter, thick filaments than the fibers of the octopus arm and have ten times as many elements (sarcomeres) in series per unit length. This gives tentacles ten times the maximum shortening velocity of arm fibers, as the maximum shortening speed of a muscle is proportional to the number of sarcomeres in series. It is, therefore, important to have more detailed information on the length and number of sarcomeres of muscle cells in various cephalopod limbs (for modeling of sarcomere ultrastructure and function in tentacles, see Van Leeuwen and Kier, 1997). Kier and Curtin estimated a maximum shortening velocity of transverse muscles in the range of 15.4 L₀ s⁻¹ in tentacles and 1.5 L₀ s⁻¹ in arms (Kier and Curtin, 2002), where L₀ is the optimal length. This difference in response is compatible with the functional roles of squid tentacles versus the squid arm, the former used for an all-or-none fast prey-capture strike and the latter in precisely modulated bending with a large range of motion (Kier and Smith, 1985). Longitudinal and transverse muscles of octopus arms manifest a mean shortening velocity of 0.356 and 0.913 L₀ s^–1, respectively (Zullo et al., 2022). Neither the transverse nor longitudinal muscles are involved in very fast motions, and the difference between them may underlie the use of longitudinal muscles in more graded responses, such as those involved in precise actions, versus the transverse muscles, which may mainly be involved in intense, sustained actions, such as maintaining posture and motion stabilization.

The control of force–length and force–velocity characteristics of agonist/antagonist muscles is fundamental for limb posture and movement. The motor system can control the lengths at which muscles start to generate force through both feedforward commands and a feedback loop between stretch receptors and agonist/antagonist muscles (for further discussion, see below).

Another puzzle is the arm's connective tissue. We may imagine muscle cells as arranged in series and in parallel, connected through a complex matrix of extracellular connective elements. Connective tissue plays a crucial role in the support and function of the arm muscles, since it can store and release elastic energy and its stiffness helps stabilize the hydrostatic limbs. A class of ‘super-coiled’ elastic fibers has been observed in the connective matrix of both the mantle mass of the squid and in cephalopod arms (Di Clemente et al., 2021; Johnsen and Kier, 1993; Kier, 2012; Kier and Stella, 2007; Kurth et al., 2014; Thompson et al., 2014; Thompson and Kier, 2001) (Fig. 1B). In longitudinal muscles they are mostly aligned in the direction of the main muscle force vector (Fig. 1B), while in transverse muscles they show a large range of orientations (Fig. 1C). This difference reflects the significantly higher stiffness of transverse versus longitudinal muscles (Di Clemente et al., 2021) as shown by experimentally derived length–tension curves (Fig. 2), where the higher passive force of the transverse muscle at the optimal length (Fig. 2B, arrowed) is indicative of its higher stiffness than in the longitudinal muscles (Fig. 2A, arrowed). As discussed below, passive and active visco-elastic properties of muscles and connective tissue contribute to the control of arm impedance (i.e. stiffness, viscosity and inertia) and are important components in controlling arm dynamics. These mechanical properties have not been sufficiently characterized in experimental and modeling studies of arm movements.

The neuromuscular system of the octopus arm consists of a packed array of mononucleated muscle cells, each receiving input from two slow and one fast motoneuron terminating at a single eye-shaped area at the center of the long (∼0.9 mm) and narrow (5–10 μm) muscle cells (Nesher et al., 2019). The small area innervated by each motoneuron encompasses around 0.2 mm³ of the arm's muscle volume (Feinstein et al., 2011; for diagrams see, Nesher et al., 2020; Zullo et al., 2019). A fast calcium action potential (together with intracellular calcium) mediates cell contraction. In contrast, in the squid transverse muscles graded sodium-based action potentials achieve maximal activation, possibly responsible for triggering and generating the rapid tentacle strikes (Gilly et al., 2020).

Synaptic inputs to the octopus arm muscles lack both the short-term synaptic plasticity and postsynaptic inhibition (Matzner et al., 2000) frequently found in other invertebrates (Atwood and Karunanithi, 2002; Bullock and Horridge, 1965). This indicates that the neuromuscular system of the octopus implements a nearly linear transformation of neuronal motor activities into muscular action similar to vertebrate neuromuscular junctions. While this organization may fit the feedforward mode of control underlying octopus reaching movements (see below; Nesher et al., 2020; Yekutieli et al., 2007), the feedforward control scheme is insufficient to describe the generation of other motions like fetching and walking. These may involve a continuous mode of contraction control integrating both feedforward- and feedback-based motor commands. Moreover, feedforward control may be insufficient to account for the dynamic tuning of muscle viscoelastic properties, not only at movement initiation but also throughout the movement.

Octopus arms both sense the environment and move. It is not clear yet how sensory inputs of different modalities provide information and if this is used locally and/or centrally to control and coordinate arm motions (Kuuspalu et al., 2022). Investigating what parameters are encoded by these inputs remains a topic of interest (Nixon and Young, 2003; Wells, 1978; Wells and Wells, 1957; Gutnick et al., 2011, 2020, 2022).

Understanding how sensory and motor areas are organized within the octopus brain is necessary for studying the sensory-motor integration leading to generating motor commands. Sensory-motor integration may be easily achieved in a somatotopically organized system. The somato-sensory body map in humans, for example, can be visualized as a distorted human-like figure – the homunculus – where the size of the cortical area dedicated to the motor or sensory functions of a body part is positively related to the significance of that part (Beisteiner et al., 2004; Huang and Sereno, 2018; Pandya et al., 2015; Penfield and Rasmussen, 1950; Sanes and Schieber, 2001; Zeharia et al., 2012).

Soft and flexible hydrostatic structures like the octopus body and limbs pose major difficulties for sensorimotor integration; it is hard to imagine how octopus motor control could be based on a somatotopic representation. This raises questions about possible mechanisms evolved in the octopus to cope with the flexibility and the kinematic and muscular hyper-redundancies characterizing the octopus body and movements.

Cephalopod brain anatomy has been extensively studied (Bidel et al., 2022 preprint; Budelmann et al., 1995; Chung et al., 2020, 2022; Deryckere et al., 2021; Maiole et al., 2019; Shigeno et al., 2018; Young, 1971; for control diagrams, see Hochner, 2012; Plan, 1987) and its functional neuronal pathways investigated with different kinds of brain preparation and techniques ranging from brain slices to in vivo recording from the central nervous system (CNS) (Brown et al., 2006; Budelmann et al., 1995; Bullock, 1984; Bullock and Budelmann, 1991; Butler-Struben et al., 2018; Sanders and Young, 1940; Shomrat et al., 2008, 2011, 2015; Wells, 1978; Gutnick et al., 2023; Zullo, 2004).

Searching for an ‘octopunculus’ (a point-to-point correspondence of octopus brain and body areas), researchers found no somatotopic organization sensu stricto within the higher motor centers of the octopus brain (Tytell, 2010; Zullo et al., 2009). Stimulation of brain areas distributed over wide regions of the higher motor centers induced both discrete and complex motions and several ‘motor primitives’, such as arm extension and crawling. The kinematic features of the evoked extensions resembled those of normal movements. Moreover, some of the movement features correlated with the stimulation parameters, for example, in arm extension, the peak velocities correlated positively with the stimulus frequencies. This suggests that the CNS may set the level of arm muscle activation (peak velocity) for the generation of a feedforward program embedded in the arm neuromuscular system (Zullo et al., 2009). This is consistent with the assumptions in modeling studies (Yekutieli et al., 2005a,b, 2007).

Even more striking is that both the motor primitives and more complex motions triggered by electrical stimulation of the octopus brain have no topographical organization. Basic and frequently used motions, such as arm extension, are represented redundantly throughout the higher motor areas and could be elicited by stimulating almost any area within them (Zullo et al., 2009).

Similarity in monkey intracortical microstimulation of motor cortex elicited visible movement or recordable electromyographic activity (EMG) (Sanes and Schieber, 2001; Schieber, 2001). Movements of a given part of the upper limb could be evoked at multiple foci scattered over a considerable portion of the representation of the upper extremity. These foci were intermingled with points whose stimulation elicited movement of other parts of the limb. More frequently used muscle combinations were represented at more locations than less frequently used muscle combinations (Gould et al., 1986; Graziano et al., 2002, 2005; Sanes and Schieber, 2001; Schieber, 2001). Schieber thus argued for a shift in the concept of somatotopy from the representation of body areas to representation of ‘actions’.

A recent concept influencing motor control research is compositionality (Abeles et al., 2013; Bizzi et al., 1991; d'Avella et al., 2015), which proposes that nervous systems construct complex motor behaviors by combining neuronally represented basic motor primitives. Each motor primitive is a computation element in the sensory-motor map that transforms desired limb trajectories into motor commands (Chiovetto et al., 2022; Flash and Bizzi, 2016; Flash and Hochner, 2005; Flash and Sejnowski, 2001; Ijspeert et al., 2013; Mussa-Ivaldi and Solla, 2004; Thoroughman and Shadmehr, 2000).

Looking for regularities in motor behavior may reveal how a particular behavior is neuronally represented or coded (Bernshteĭn, 1967; Bizzi et al., 1991; Georgopoulos et al., 1982; Mussa–Ivaldi and Bizzi, 2000). This approach has been applied in vertebrates at the neural, muscular, joint and end-effector levels (D'Avella and Lacquaniti, 2013) and has emphasized the importance of considering not only kinematic and dynamic aspects of unconstrained motions but also the control of limb impedance, especially during interaction tasks (Hogan and Sternad, 2012, 2013). Recordings from large neural populations in monkeys (Churchland et al., 2012; Hatsopoulos et al., 2007) and other animals allowed efforts to identify both the basic motor units and the grammatical rules subserving generation of a large repertoire of motor actions by combining a limited number of motor building blocks (Chiovetto et al., 2022; d'Avella et al., 2015; Flash and Bizzi, 2016; Flash and Hochner, 2005; Ijspeert et al., 2013; Thoroughman and Shadmehr, 2000).

To examine the relevance of motor compositionality for cephalopod motor control, researchers kinematically analyzed several basic movements frequently used in various actions, such as arm bending, extension, shortening-elongation and torsion movements. A computational study of octopus arm extensions demonstrated that these can be constructed from a few elementary motion primitives (Zelman et al., 2013). More complex movements such as crawling, walking, fetching and swimming were also shown to be composed of basic movements (described below) (Gutfreund et al., 1996; Hanassy et al., 2015; Levy et al., 2015; Richter et al., 2015; Sumbre et al., 2006). Overall, these observations suggest the existence and use of motion primitives, although these are not somatotopically represented centrally. However, the sequences of contraction and relaxation of different muscle types have not yet been studied and therefore their functional roles in generating specific motions had to be inferred mostly based on their orientations and arrangement within the arm (Kier and Thompson, 2003; Kier, 2016; Kier and Stella, 2007). In comparison, combined kinematic and EMG recordings in elephant trunks showed strong associations between motion primitives and muscular synergies, where trunk curvature, torsion and strain provide an appropriate kinematic and kinetic characterization of the trunk trajectories and motion primitives (Dagenais et al., 2021; Hooper, 2021).

Several basic motion primitives have been inferred from video recordings of movement and kinematic analysis of octopus arm movements (Fig. 3). Reaching movements involving arm extension are observed in many octopus behaviors, such as reaching to grasp an object (Gutfreund et al., 1996), two-legged locomotion (Huffard et al., 2005) and searching motions. Octopuses reach toward a target in a stereotyped manner using a basic invariant motor structure: a bend forming at the base (or any region) of the arm and then travelling toward the tip (Gutfreund et al., 1996). To study the neuronal control of these movements, arm muscle activations were measured simultaneously with the kinematics. Bends can be formed by activation of longitudinal muscles with the concomitant resistance to shortening produced by the activation of any of its antagonistic muscles (transverse, radial or oblique) (Kier and Smith, 1985). Bending can be also favored by passive forces intrinsic to the octopus arm, such as the higher stiffness of transverse muscles (thus resisting compression) and the higher elasticity of longitudinal muscles (Di Clemente et al., 2021). Once formed, the bend travels along the arm by propagating a wave of muscle activation, with peak muscle activity slightly preceding the traveling bend. EMG activity during the initial stages of movement predict the peak velocity attained toward its end. Thus, feedforward motor commands are important for control of movement velocity and simple adjustments of the excitation levels at the onset of movement can set the velocity profile of the whole movement. Bend production and then propagation are due to specific and well-coordinated contractions of the transverse and longitudinal muscles. The arm segment proximal to the bend remains stiff as the bend propagates towards the tip (Gutfreund et al., 1996, 1998).

Arm extension can be reproduced both through in vivo brain microstimulation in freely behaving animals (Zullo et al., 2009) and ex vivo in an isolated arm (Sumbre et al., 2001). This finding supports production of the reaching motion involving a feedforward control scheme based on peripherally generated neural commands leading to synergistic muscle activations and dictating the kinematic characteristics of the reaching movement.

Another basic motor behavior in the octopus is arm elongation. In muscular hydrostats, elongation is generated through the contraction of muscles decreasing the cross-sectional area of the organ (transverse, oblique and radial muscles) (Kier and Smith, 1985). In the octopus, arm elongation can be generated by contraction of the transverse and circular muscles resulting in passive elongation of the longitudinal muscles. Radial muscles play a major role in squid arm movements and are involved in generating elongation movements in other muscular hydrostats such as the elephant trunk and the tongue.

Hanassy et al. (2015) noticed that during octopus arm reaching behavior, elongation of the arm segment proximal to the bend point occurred simultaneously with bend propagation. The elongations were substantial and highly variable. Reaching movements can be broadly segregated into two groups. One involves bend propagation from the base of the arm towards its tip, and in the other, the bend is formed or present more distally with reaching achieved mainly by elongation and straightening of the segment proximal to the bend. Only in the second type of movement does the arm elongation positively correlate with the distance of the bend from the target (Hanassy et al., 2015). Thus, reaching a target appears to be generated by a combination of both bend propagation and arm elongation. These two motor primitives may be combined to create a broad spectrum of reaching movements.

Octopuses show a large variety of torsion movements (Kennedy et al., 2020). They are generated by three pairs of oblique muscles (Kier, 2016), two of which are arranged in both right- and left-handed helices. The external and medial oblique muscles on each side of the arm together with the cross-fiber connective tissue array represent both left and right helical systems allowing torsion forces to be generated in both directions. Co-contraction of both pairs increases arm stiffness. The third pair of oblique muscles are situated closer to the neural cord, possibly protecting its integrity. The orientation of the helical muscle with respect to the arm's long axis determines whether their contractions contribute forces longitudinally or transversely along the arm. A helical fiber is shortest when the fiber angle equals 54 deg 44′ and longest when the fiber angle approaches 0 deg and 90 deg. That is, when helically arranged muscle fibers with fiber angle greater than 54 deg 44′ contract, they create a force for both torsion and elongation. Helically arranged muscle fibers with a fiber angle of less than 54 deg 44′ create a force for both torsion and shortening along the main or transverse axis of the arm. The angle at which they contribute equal forces in both directions is approximately 54 deg (Kier and Thompson, 2003; Kier, 2016; Kier and Smith, 1985; Kier and Stella, 2007).

Stiffening is an important component of various movements in muscular hydrostats. It can be produced by multi-axial contractions involving the co-activation of antagonistic muscles and by muscle contractions passively resisted by connective tissue (Kier and Smith, 1985; Kier and Stella, 2007). Connective tissue provides structural reinforcement during muscle stiffening and plays a critical role in controlling the shape and motions of the muscular hydrostat, which when stiffened, may be used as supportive elements, such as during arm fetching movements and walking (Kier and Smith, 1985; Huffard et al., 2005; Sumbre et al., 2006).

Compositionality suggests that complex motions derive from combining a limited set of basic motions (Abeles et al., 2013). In addition to the basic motion primitives described above, the octopus also manifests rhythmic behaviors, such as walking (Huffard et al., 2005), crawling (Levy et al., 2015) and swimming (Kazakidi et al., 2012), and movements generated by the interplay between sensory stimuli and motor commands, such as fetching.

Octopuses fetch food to their mouth using a stereotypic reconfiguration of the arm. When a piece of fish is presented to a freely moving octopus, it extends its arm towards the fish, grabbing it with the distal part of its arm. It then brings the food towards its mouth by rotating the distal half of the arm around a pseudo-joint created midway between the distal part and the base of the arm. EMG recordings mostly from longitudinal and transverse muscles detected two counter-propagating waves of muscle activity positively correlating with the segment lengths and with the site on the arm at which the food was grasped (Sumbre et al., 2006).

The ingenious mechanism of counter-propagating waves has also been observed in the electric fish and lamprey, where a central pattern generator (CPG) model for single-traveling wave kinematics showed gait transitions between forward waves, reverse waves, asymmetrical waves and varying wavelengths (Zhou and Low, 2010, 2012). A similar CPG model adapted from the lamprey model reproduced the kinematics of both single traveling and counter-propagating waves in the electric knifefish ribbon fin (Ruiz-Torres et al., 2013).

These observations suggest two alternative hypotheses for the neural control of octopus fetching movements. In a ‘labeled line’ signal conductance model, the brain activates a peripheral program at a specific location along the arm through axonal bundles specifically innervating various segments of the arm neuromuscular system. In the alternative control hypothesis, this behavior is generated through a deformation-based mechanism in the distal joint caused by proprioceptive and/or sensory waves of activation triggered by the arm's contact with the food and transmitted along the arm (Zullo et al., 2011). Testing these two hypotheses, Zullo and colleagues rejected the labeled-line hypothesis since there is no indication of axonal tracts allowing the brain to activate a peripheral program at a specific location along the arm. Rather, the axonal bundles seem to innervate the motor neuron pools along the arm en passant, so that axons may activate long muscle sections. In this scenario, to create a bend or stiffness in a confined area, centrally originating motor commands have to be integrated at such locations with local sensory signals from the suckers and skin and proprioceptive signals (Zullo et al., 2019).

A partial solution to the puzzle of pseudo-joint generation during fetching movements was provided by Sumbre et al. (2001, 2005, 2006), who showed that unlike bending movements, the fetching movement could not be generated by simple activation of a peripheral motor input. Generating a fetching movement may require tight temporal coupling between centrally triggered muscle activations moving rostro-caudally and muscular activities moving in the opposite direction possibly triggered by sensory feedback from chemical or mechanical inputs generated through the arm contact with the grasped food. And indeed, arm sensory responses do appear to be an integral part of the peripheral neural responses allowing the generation of stereotypical motions (Gutfreund et al., 1996, 1998, 2006).

Rostro-caudal or caudo-rostral waves were also observed in the lamprey (Matsushima and Grillner, 1990, 1992) and salamander (Ijspeert et al., 2013). Propagation of an undulatory movement may be achieved according to the ‘trailing oscillator’ hypothesis, whereby activating a leading oscillator more strongly than the consecutive oscillator may cause a wave to move from the leading oscillator forward (Matsushima and Grillner, 1990). A reverse propagating wave will move from the more distal toward the more proximal part of the lamprey's body if the more strongly activated oscillator is located at the distal part of the animal's body. Whether a similar mechanism operates in the octopus is still an open question but, if so, this would be a simple means for reproducing the control, mechanics and forces acting during the generation of arm fetching movements.

Using a fast Fourier transform analysis, Levy et al. (2015) found no evidence of rhythmicity in arm coordination during octopus crawling. Thus, crawling behavior must involve different mechanisms of movement generation from those associated with conventional CPGs which drive the rhythmicity usually observed in locomotion (Levy et al., 2015). However, further evidence is still needed to accept or reject the trailing oscillator hypothesis (Matsushima and Grillner, 1990) for non-periodic motions such as fetching and elongation.

The development of sophisticated brain and muscle imaging techniques in mammals has allowed considerable progress in identifying motion building blocks (e.g. muscle synergies or local neural modules). Experimental methods, however, are not enough to advance our understanding of the hierarchical and modular organization of the motor system. These must be augmented by mathematical and biomechanical modeling. In invertebrates, particularly in cephalopods, owing to the inherent technical difficulties and the relative slowness in developing neural imaging techniques, new insights into the organization of the neuromuscular control systems have been enabled by the combination of modeling with experimental studies.

Since early dynamic models were based on the limited biomechanical and neural control knowledge available at the time, we discuss discrepancies between the predictions of these models and various characterized kinematic features of different types of octopus movements. These discrepancies have raised questions of the validity of the force–length characteristics and muscle viscoelastic properties used in these earlier dynamic models. We will illustrate how recently updated models of muscle force generation may resolve some of the current puzzles of motor control in muscular hydrostats and the possible contributions that these updated models may make to the design and control of soft robotic arms.

The first neuromuscular control model developed by Van Leeuwen and Kier presented a detailed quantitative analysis of tentacle extension in squids by integrating across several levels of structural organization (Van Leeuwen and Kier, 1997). Motion capture analysis of squid tentacle extensions showed that their muscular stalks rapidly elongate by 70% of their resting length during prey capture. In their forward dynamics model of this extension, the tentacular stalk was modelled as a longitudinal array of thin muscular discs with extensor muscles oriented parallel to the disc planes. As a disc contracted radially, its constant volume forced it to lengthen. The inputs to the model were the tentacle dimensions, passive and active muscle properties described by the Hill's force–velocity relationship, myo-filament lengths and activation of the muscles. The model predicted the changing geometry of the tentacles, the pressure and stress distribution inside the tentacle and the velocity and kinetic energy distribution of the stalk and club. These predictions fitted kinematic observations from high-speed films of prey capture, demonstrating that the unusually short myosin filaments (0.5±0.9 µm) characterizing the extensor muscles are necessary for the very rapid tentacle extension. Additionally, the model allowed exploration of the effects of changes in the dimensions and mass of the tentacle and intrinsic speed of the myofilaments on the optimum myosin filament lengths.

More recently, Yekutieli et al.'s 2D dynamic model of the octopus arm explored octopus neuromuscular control strategies (Yekutieli et al., 2005a,b, 2007). Following Van Leeuwen and Kier (1997), the octopus arm was modeled as a multi-segment structure (Fig. 4A), each segment containing longitudinal and transverse muscles and maintaining a constant volume (Fig. 4B). The input to the model was the degree of activation of each of its muscles. The model included the experimentally estimated external forces of gravity, buoyancy and water drag. It also included the internal forces generated by the arm muscles and the forces responsible for maintaining a constant volume that give rise to the internal pressure of the arm segments. The arm was modeled using a finite-element-like model with each segment controlled by two longitudinal and two transverse muscles (Fig. 4A). The muscle forces were based on Hill's model resulting from a multiplication of force–length (Fig. 4B), force–velocity and force–frequency mathematical functions (see also Ting and Chiel, 2017) describing known viscoelastic properties of activated muscles but with fixed rest lengths and with similar mechanical properties for the transverse and longitudinal muscles.

The dynamic model was used to investigate the arm movements of an octopus reaching toward an object (Fig. 4C). Testing alternative schemes of neuromuscular control, the model predicted: (1) a simple command producing a wave of muscle activation moving at a constant velocity is sufficient to replicate the natural reaching movements with similar kinematic features; (2) the reaching movement is produced by a stiffening wave of muscle contraction that pushes a bend forward along the arm; and (3) muscles employing either linear or nonlinear force–length relationships produced similar simulated movements.

Note, however, the new evidence that the transverse and longitudinal muscles of the octopus arm differ in their force–velocity and force–frequency responses (Zullo et al., 2022). Transverse muscles have lower contraction and shortening velocities and forces increase more steeply at lower frequencies of stimulation. In addition, the ratio of the arm's volume occupied by transverse and longitudinal muscles differs between the oral and aboral arm regions, and their volume ratio varies along the arm (Zullo et al., 2022). This suggests that these muscles manifest different mechanical characteristics, requiring revision of the model.

Other modeling studies have extended the dynamic models, representing the arm as 3D continuous structures (Godage et al., 2011a; Guglielmino et al., 2010) using continuous models of the arm (Ian et al., 2005) and variable viscous-elastic properties (Renda et al., 2014). Yet others have incorporated mathematical descriptions of hydrodynamic forces that become particularly important in simulating octopus swimming (Kazakidi et al., 2012). Fluid–structure interactions (FSIs) are physical phenomena occurring when a fluid exerts forces on a solid structure, which can deform or move the structure. This deformation or movement, in turn, affects the behavior of the fluid. Cephalopods use FSIs to control their movement and shape (Wang et al., 2022a); for example, in squids, the muscular mantle surrounding the animal's body generates water jets that propel the animal forward (Bartol et al., 2009). The shape of the mantle and the direction and timing of the jets can be adjusted to produce different patterns of movement (Zhu and Xiao, 2022). Understanding the mechanisms of FSI in cephalopods may have important implications for the design of underwater vehicles and biomimetic materials. Cephalopod swimming studies have inspired design and development of control schemes for soft robotic swimmers (Sfakiotakis et al., 2013, 2015). Some studies have focused on designing and building soft-robotic octopus-like manipulators (Laschi et al., 2009, 2012, 2016; Margheri et al., 2012; Mazzolai et al., 2012; Trivedi et al., 2008).

While many of these soft robotic arms were biologically inspired (e.g. Rus and Tolley, 2015), other studies focused on developing mathematical models and control schemes and using computer simulations to test whether the predicted motions successfully replicated the kinematic characteristics of basic octopus motions (bending, arm elongation, fetching and torsion). Noteworthy are recent studies in which the Cosserat rod theory was used to model movements of soft biological and robotic arms. Cosserat rod models describe the behavior of slender elastic structures, such as rods, fibers and filaments (Gazzola et al., 2018), and they consider not only the deformation of such structures but also their rotation and twisting. The elastic rod is represented as a collection of cross-sectional elements whose deformations are described using a set of equations relating the forces and moments acting on those elements to their deformation and rotation. In several recent studies, these models were applied to the movements of the octopus arm during reaching, fetching and grasping. The model accounted for the elastic properties of the arm and the forces generated by its muscular contractions. By simulating these movements, the researchers were able to gain insight into the underlying mechanisms of octopus movement, model neuromuscular control and sensory feedback (Wang et al., 2022b preprint; Wang et al., 2022c) and identify the most efficient movements for different tasks using the energy-shaping approach (Chang et al., 2020a,b; Zhang et al., 2019).

However, many of the recently developed dynamic models lack realistic representations of the detailed anatomy of the octopus. Moreover, they use biomechanical models that do not include sufficiently realistic or updated biomechanical representations (e.g. Nishikawa, 2020). Nor do they model motor control as having both local and global effects including tuning of the body and muscle force, stiffness and viscosity.

For example, Hanassy et al. (2015) used the model of Yekutieli et al. (2005a,b, 2007) to examine which neural control strategies can account for the generation of arm elongation. However, this model required unrealistically high ratios of transverse to longitudinal muscle activation to successfully replicate observed elongation movements.

Although recent studies on isolated arm muscles showed that these muscles do not significantly differ in their active length–tension relationships, in vivo under natural anatomical configurations, the transverse and longitudinal muscles experience different stresses and operate within different ranges of their length–tension curves, thus producing different amounts of force when activated (Di Clemente et al., 2021; Zullo et al., 2022) (Fig. 2).

That neural activation may have different effects on the transverse and longitudinal muscle rest-lengths and viscoelastic properties suggests possible remedies for overcoming the unrealistic high ratios of transverse to longitudinal neural activations required to account for experimentally observed elongation movements.

In the earlier models of arm dynamics and control of the octopus arm (e.g. Godage et al., 2011a,b; Guglielmino et al., 2010; Van Leeuwen and Kier, 1997; Yekutieli et al., 2005a,b, 2007; Zheng et al., 2012), both passively and actively generated forces were based on Hill's model. Additionally, force–length and force–velocity properties were assumed to emerge from behaviors obeying the cross-bridge and sliding filament models. The parameters used to model the force–length and force–velocity characteristics were taken from isometric and isotonic experimental measurements of muscle biomechanical relationships. In modeling the morphological properties of the arm, the obliquely striated structure of the longitudinal and transverse muscles was not considered. The passive and active viscoelastic properties of the muscles (stiffness, damping) and limb reflected these underlying structural and biomechanical assumptions.

The earlier, highly dominant cross-bridge and sliding filament models have now been updated, and new experimental studies have tested the validity of the new models of muscle biomechanics. Of great importance is the awareness that the huge protein titin within the sarcomeres affects the length–tension and length–velocity properties. Titin is connected in series with the myosin filaments extending between the Z- and M-lines in the sarcomere (Fig. 5A). The titin protein behaves like a tunable spring affecting the passive stiffness of the sarcomeres and muscle fibers (and their length–velocity characteristics). Even more importantly, titin may influence the length-dependent activation of muscles (Herzog, 2018; Herzog et al., 2016; Hessel et al., 2019; Nishikawa, 2020; Tahir et al., 2020) by varying their activation response and frequency-dependent modifications of the length–tension properties, similarly to the force–frequency dependency in electrically stimulated muscles (Fig. 5B) (Brown et al., 1999; Fukuda et al., 2001, 2003; Malamud, 1989; Rack and Westbury, 1969; Roszek et al., 1994; Terui et al., 2008. For recent findings in this highly active research area see Herzog, 2018; Herzog et al., 2016; Monroy et al., 2012; Nishikawa, 2020).

It is now clear that the sliding filament–cross bridge theory fails to predict skeletal muscle forces during dynamic length changes (Heidlauf et al., 2017; Hessel et al., 2021). Since these changes are extreme in hydrostatic bodies and may occur in a fast and continuous manner during motion, it is fundamental to further investigate the role of viscoelastic elements, such as titin, during dynamic length changes in cephalopod muscles. Also needing examination is the interaction between the obliquely striated muscular geometry and the effect of titin on muscle visco-elastic properties. Most of the experimental studies on the effect of titin on these properties have used striated skeletal muscles (Herzog, 2018; Nishikawa, 2020), and more recently, squid muscles (Taylor-Burt et al., 2018; Williams and Holt, 2018). As yet, the existence of titin has not been confirmed in octopus arm muscles; however, a titin-like giant thick-filament protein, twitchin, is found in molluscan catch muscles (Funabara et al., 2005). Also, both titin-like proteins and twitchin are present in the octopus genome and transcriptome. Thus, it will be pertinent to assess their expression specifically in muscular hydrostat tissues.

Besides titin-related effects on muscle stiffness and rest length, other mechanisms play a role in tuning muscle force–length and force–velocity relationships. As shown recently, activation can affect the static and dynamic mechanical properties of squid cross-striated muscles. Studies of squid mantle cross-striated muscle fibers have demonstrated that the level of activation can affect the slope (stiffness) of the ascending limb of the length–tension relationship (Thompson et al., 2014), similarly to observations in vertebrate skeletal muscles (Rack and Westbury, 1969). In obliquely striated squid muscles, the maximum isometric twitch force is produced at a shorter length than the length of the isometric tetanic force (Thompson et al., 2014). This led to the hypothesis that obliquely striated muscles operate over an extreme range of muscle lengths, much greater than those of cross-striated fibers. Recent studies also showed that the initial striation angle has a large effect on the muscle force–length relationship (Taylor-Burt et al., 2018; Thompson et al., 2023).

As octopus transverse and longitudinal muscles serve as agonist/antagonist pairs in many different tasks, their rest-lengths may also be actively controlled. In addition, it is important to measure impedance (stiffness and viscosity) and muscle rest-length tuning in the octopus, given that these properties are neuronally controlled and play significant roles in both unconstrained motions and interaction tasks (Hogan, 1985; Hogan and Buerger, 2005). However, because of the difficulties associated with measuring stiffness, especially in actively moving animals, one may need to resort to muscle models, inferring the limb impedance through computer simulations (e.g. Held et al., 2012).

Future studies should combine biological experiments with modeling studies to measure the neural control of muscle rest lengths, stiffness, and velocity-dependent forces and the mechanisms underlying the tuning of these variables during posture and movement. Hopefully, such studies will allow us to examine whether the arm deformations produced can be linked to sarcomeres stiffness and stretch-related effects on the length–velocity mode of operation. Further work is also needed to clarify whether the frequency of activation and the number of muscle fibers recruited affect muscle rest-length and stiffness. In future modeling of the octopus arm, the mechanical and neural control factors discussed above should be considered, including the ‘non-muscular’ components of the extracellular matrix. Such studies will provide insights into central and distributed neural control of biological muscular hydrostats that are also important for the field of biologically inspired robotics, especially by leveraging on the morphological structure and mechanical properties of biological actuators (Higueras-Ruiz et al., 2022).

The concepts of embodiment and morphological computing may be useful in reconciling the high kinematic and muscular redundancy and autonomy of the octopus arm with the absence of a topographic brain representation. Indeed, these concepts suggest that the morphology and organization of the octopus body and nervous system work synergistically to control motion without a body representation (for reviews, see Chiel and Beer, 1997; Chiel et al., 2009; Hochner, 2012, 2013; Levy and Hochner, 2017; Zullo and Hochner, 2011). In cephalopods, performing various motion primitives involves synergistic control of different muscle groups. Embodied organization has been stressed both in robotics research and biological contexts. Advancing our understanding of its implementation in soft biological arms, tentacles and bodies may provide novel insights for new engineering approaches based on adaptability of body morphology and dynamic control of the behavioral tasks the robot is required to perform (Brooks, 1991a,b;Jordanous, 2020; Kang et al., 2016; Nakajima et al., 2013; Pfeifer et al., 2007, 2014; Zullo et al., 2012). This is certainly very important when dealing with autonomous robots, which need to adjust autonomously to the environment in which they operate. Using an embodied organization for the control of soft robots will greatly reduce the computational complexity and allow a higher degree of autonomy than currently achieved.

The research on the biomechanics and motor control of octopuses and other muscular hydrostats reviewed here coincides with an emphasis on the development of soft robotic systems, many inspired by the insights from research on muscular hydrostats. Advancing both fields requires a deeper understanding of the basic principles involved in controlling muscle forces, both under passive and active conditions, as well as during the control of posture and dynamic tasks. Other important directions are the neural control of basic motions and the control of muscle synergies through combining different elementary movements and executing them by temporally and spatially coordinating the activities of different muscle groups. Such efforts can be supported through developing computational models and examining alternative control principles and strategies, also considering the distributed neural sensory and motor representations in octopus brain revealed by microstimulation. This novel understanding of biomechanics, morphology and neural motor control of soft limbs and bodies can lead to the development of new actuators and novel control schemes for soft robotic systems, reducing the computational complexities associated with movement generation by artificial soft limbs.

We thank Alessio di Clemente and Federica Maiole for the octopus biomechanics and imaging work described in this paper. We thank Prof. Jenny Kien for valuable suggestions and editing of this manuscript.